I have a target that I shot with a variety of cast and jacketed bullets and all had more or less the same mean point of impact.
Rifles that shoot different weight bullets to the same point of impact are scarcer than hen's teeth. I have only encountered one in my lifetime that would, and I still own it. It is a Ruger #1V chambered for 7mm Rem. Mag. I have shot bullet weights from 115 grs. to 145 grs. in it and they need no scope adjustment nor Kentucky windage to be accurate to POA. I cannot understand or explain the phenomenon but, it's pretty cool to see.